This is pretty much my take on the whole thing, although I actually don't know the debate in detail (something I know I should clarify now before someone like Robert Vienneau comes by and challenges me on it):
"“Cambridge UK” (i.e. Joan Robinson) believed that this destroys
neoclassical economics and that only Sraffian economics had a way to
explain interest rates correctly. This is because, if you present
entrepreneurs as deciding between two production methods (one
labor-intensive and the other capital-intensive), they may sometimes
find it profit-maximizing to switch back and forth between the capital intensity as the interest rate shifts.
This means that there is more than one interest rate corresponding to
an allocation of capital, in turn meaning that the interest rate is
determined by arbitrary institutions, not the logic of marginal
analysis. Of course, this is all an artifact of the algebraic
formalization of the problem in the first place; the same mathematical issue comes up if you try using things like internal rate of return.
The response of people who know cost-benefit analysis is that you
shouldn’t use internal rate of return. Similarly, most people would see
this as demonstrating that you shouldn’t take the concept of marginal
efficiency of capital too seriously. “Cambridge US,” despite its
infatuation with formalistic Samuelsonian methodological descriptivism
(say that five times fast), kinda understood that such modeled results
are cute but not particularly meaningful. “Cambridge UK,” because it was
a group of Marxists, was giddy to find any excuse to say that interest
rates are too high and the system is unjust. Which is why they still
won’t shut up about it and mainstream neoclassical economists don’t care
about it." (emphasis added)
The bit about Marxism is particularly insightful. If you ever get the chance to read a little Marxist economics (as opposed to some other Marxist work), it's incredible how riled up they get over a little algebra. Actually it's reminiscent of a lot of the MMT people, with one exception: MMTers actually do work with data a lot and get into some behavioral discussions around what drives financial markets. In other words, they check the data and think more about microfoundations and don't just take some algebra for granted. The theory is (predictably) more useful as a result.
I think Samuelson just liked playing around with the math, because he got involved in the transformation problem and the Marxists too.
I should probably stop here because as I said - I'm only vaguely familiar with all of this. But Ryan's response rang true for me. One bit of Marx I have read in much more detail is his work on Say's Law. That is actually quite good, and in a lot of ways better than Malthus's writing on it (which is also quite good).
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